Motivation: Call the above sum $S$, and let
$$T := \sum_{ GCD(i,j)=1} \frac{1}{\max(i,j) i j}.$$
The sum $T$ came up in a computation on Jim Propp's question here. Numerical computation suggested that $T$ is extremely close to $3$.

This question appears to be related to a recent one of mine: mathoverflow.net/questions/50253. I'm wondering if one of the solutions given there can be used to give another perspective on this result?
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Mike SpiveyFeb 11 '11 at 19:26

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@Mariano: aside from the fact that I remembered seeing the identity on that page a few years back, finding it again was routine: google "Riemann Zeta Function", second hit, skimming to about mid page, and there, but not over - first reference Stark, jstor, and he references Klamkin, as did David below. As can be seen from the timestamps, this was done in under 5 minutes (given that I remembered seeing it).
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Dror SpeiserFeb 16 '11 at 10:14

Thanks Marty! For those with JSTOR access, a very clean proof is given at jstor.org/stable/2308345 . For those with access to a good library, this is The American Mathematical Monthly, Vol. 59, No. 7, Aug. - Sep., 1952, p. 471 , problem proposed by M. Klamkin, solution by R. Steinberg.
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David SpeyerFeb 11 '11 at 17:48